On the performance of a cavity method based algorithm for the Prize-Collecting Steiner Tree Problem on graphs

We study the behavior of an algorithm derived from the cavity method for the Prize-Collecting Steiner Tree (PCST) problem on graphs. The algorithm is based on the zero temperature limit of the cavity equations and as such is formally simple (a fixed point equation resolved by iteration) and distributed (parallelizable). We provide a detailed comparison with state-of-the-art algorithms on a wide range of existing benchmarks networks and random graphs. Specifically, we consider an enhanced derivative of the Goemans-Williamson heuristics and the DHEA solver, a Branch and Cut Linear/Integer Programming based approach. The comparison shows that the cavity algorithm outperforms the two algorithms in most large instances both in running time and quality of the solution. Finally we prove a few optimality properties of the solutions provided by our algorithm, including optimality under the two post-processing procedures defined in the Goemans-Williamson derivative and global optimality in some limit cases

Approximation Algorithm for Unrooted Prize-Collecting Forest with Multiple Components and Its Application on Prize-Collecting Sweep Coverage

In this paper, we introduce a polynomial-time 2-approximation algorithm for the Unrooted Prize-Collecting Forest with $K$ Components (URPCF$_K$) problem. URPCF$_K$ aims to find a forest with exactly $K$ connected components while minimizing both the forest's weight and the penalties incurred by unspanned vertices. Unlike the rooted version RPCF$_K$, where a 2-approximation algorithm exists, solving the unrooted version by guessing roots leads to exponential time complexity for non-constant $K$. To address this challenge, we propose a rootless growing and rootless pruning algorithm. We also apply this algorithm to improve the approximation ratio for the Prize-Collecting Min-Sensor Sweep Cover problem (PCMinSSC) from 8 to 5. Keywords: approximation algorithm, prize-collecting Steiner forest, sweep cover

Stochastic Network Design: Models and Scalable Algorithms

Many natural and social phenomena occur in networks. Examples include the spread of information, ideas, and opinions through a social network, the propagation of an infectious disease among people, and the spread of species within an interconnected habitat network. The ability to modify a phenomenon towards some desired outcomes has widely recognized benefits to our society and the economy. The outcome of a phenomenon is largely determined by the topology or properties of its underlying network. A decision maker can take management actions to modify a network and, therefore, change the outcome of the phenomenon. A management action is an activity that changes the topology or other properties of a network. For example, species that live in a small area may expand their population and gradually spread into an interconnected habitat network. However, human development of various structures such as highways and factories may destroy natural habitats or block paths connecting different habitat patches, which results in a population decline. To facilitate the dispersal of species and help the population recover, artificial corridors (e.g., a wildlife highway crossing) can be built to restore connectivity of isolated habitats, and conservation areas can be established to restore historical habitats of species, both of which are examples of management actions. The set of management actions that can be taken is restricted by a budget, so we must find cost-effective allocations of limited funding resources. In the thesis, the problem of finding the (nearly) optimal set of management actions is formulated as a discrete and stochastic optimization problem. Specifically, a general decision-making framework called stochastic network design is defined to model a broad range of similar real-world problems. The framework is defined upon a stochastic network, in which edges are either present or absent with certain probabilities. It defines several metrics to measure the outcome of the underlying phenomenon and a set of management actions that modify the network or its parameters in specific ways. The goal is to select a subset of management actions, subject to a budget constraint, to maximize a specified metric. The major contribution of the thesis is to develop scalable algorithms to find high- quality solutions for different problems within the framework. In general, these problems are NP-hard, and their objective functions are neither submodular nor super-modular. Existing algorithms, such as greedy algorithms and heuristic search algorithms, either lack theoretical guarantees or have limited scalability. In the thesis, fast approximate algorithms are developed under three different settings that are gradually more general. The most restricted setting is when a network is tree-structured. For this case, fully polynomial-time approximation schemes (FPTAS) are developed using dynamic programming algorithms and rounding techniques. A more general setting is when networks are general directed graphs. We use a sampling technique to convert the original stochastic optimization problem into a deterministic optimization problem and develop a primal-dual algorithm to solve it efficiently. In the previous two problem settings, the goal is to maximize connectivity of networks. In the most general setting, the goal is to maximize the number of nodes being connected and minimize the distance between these connected nodes. For example, we do not only want the species to reach a large number of habitat areas but also want them to be able to get there within a reasonable amount of time. The scalable algorithms for this setting combine a fast primal-dual algorithm and a sampling procedure. Three real-world problems from the areas of computational sustainability and emergency response are used to evaluate these algorithms. They are the barrier removal problem aimed to determine which instream barriers to remove to help fish access their historical habitats in a river network, the spatial conservation planning problem to determine which habitat units to set as conservation areas to encourage the dispersal of endangered species in a landscape, and the pre-disaster preparation problem aimed to minimize the disruption of emergency medical services by natural disasters. In these three problems, the developed algorithms are much more scalable than the existing state-of-the-arts and produce high-quality solutions

Statistical mechanics approaches to optimization and inference

Nowadays, typical methodologies employed in statistical physics are successfully applied to a huge set of problems arising from different research fields. In this thesis I will propose several statistical mechanics based models able to deal with two types of problems: optimization and inference problems. The intrinsic difficulty that characterizes both problems is that, due to the hard combinatorial nature of optimization and inference, finding exact solutions would require hard and impractical computations. In fact, the time needed to perform these calculations, in almost all cases, scales exponentially with respect to relevant parameters of the system and thus cannot be accomplished in practice. As combinatorial optimization addresses the problem of finding a fair configuration of variables able to minimize/maximize an objective function, inference seeks a posteriori the most fair assignment of a set of variables given a partial knowledge of the system. These two problems can be re-phrased in a statistical mechanics framework where elementary components of a physical system interact according to the constraints of the original problem. The information at our disposal can be encoded in the Boltzmann distribution of the new variables which, if properly investigated, can provide the solutions to the original problems. As a consequence, the methodologies originally adopted in statistical mechanics to study and, eventually, approximate the Boltzmann distribution can be fruitfully applied for solving inference and optimization problems. The structure of the thesis follows the path covered during the three years of my Ph.D. At first, I will propose a set of combinatorial optimization problems on graphs, the Prize collecting and the Packing of Steiner trees problems. The tools used to face these hard problems rely on the zero-temperature implementation of the Belief Propagation algorithm, called Max Sum algorithm. The second set of problems proposed in this thesis falls under the name of linear estimation problems. One of them, the compressed sensing problem, will guide us in the modelling of these problems within a Bayesian framework along with the introduction of a powerful algorithm known as Expectation Propagation or Expectation Consistent in statistical physics. I will propose a similar approach to other challenging problems: the inference of metabolic fluxes, the inverse problem of the electro-encephalography and the reconstruction of tomographic images

A note on the data-driven capacity of P2P networks

We consider two capacity problems in P2P networks. In the first one, the nodes have an infinite amount of data to send and the goal is to optimally allocate their uplink bandwidths such that the demands of every peer in terms of receiving data rate are met. We solve this problem through a mapping from a node-weighted graph featuring two labels per node to a max flow problem on an edge-weighted bipartite graph. In the second problem under consideration, the resource allocation is driven by the availability of the data resource that the peers are interested in sharing. That is a node cannot allocate its uplink resources unless it has data to transmit first. The problem of uplink bandwidth allocation is then equivalent to constructing a set of directed trees in the overlay such that the number of nodes receiving the data is maximized while the uplink capacities of the peers are not exceeded. We show that the problem is NP-complete, and provide a linear programming decomposition decoupling it into a master problem and multiple slave subproblems that can be resolved in polynomial time. We also design a heuristic algorithm in order to compute a suboptimal solution in a reasonable time. This algorithm requires only a local knowledge from nodes, so it should support distributed implementations. We analyze both problems through a series of simulation experiments featuring different network sizes and network densities. On large networks, we compare our heuristic and its variants with a genetic algorithm and show that our heuristic computes the better resource allocation. On smaller networks, we contrast these performances to that of the exact algorithm and show that resource allocation fulfilling a large part of the peer can be found, even for hard configuration where no resources are in excess.Comment: 10 pages, technical report assisting a submissio